We have 2 desktop computers and we would like to connect them together in order to transfer data from one computer to another. Is it possible to connect them with a cord? Only the computer we use is connected to internet. If we can't connect them with a cord, can we physically take the harddrive out and connect it to the other computer, save the data on it, and then transfer it back?

Yes and Yes...

You can connect the two PC's via a crossover cable and then network them together to share files with each other.

You can take the hard drive out of one PC and slave it to the hard drive in the other PC and transfer files that way.

Hope this helps,
Dave02

Keep in mind that a crossover cable LOOKS just like a regular LAN cable but internally some of the wires crossover. You may have better luck finding an adapter that you plug into one of the computer networking cards, connect a standard netwoking cable (LAN) to the adapter and the other computer's networking card. You may need to change some settings on one or both of the computers, such as setting up sharing files. The above way is the simple way.

in the united kingdom where i am a 'crossover' cable costs about £2 including shipping on ebay (about 4 us dollars). As long as both computers have unused network/lan/rj45 socket on them then that is all you need to spend to have file sharing between the 2 computers and have them both sharing the internet connection.
(for internet, the 'primary' internet computer needs to be on for the 'secondary' computer to access the internet, if done on this very cheap way).
How you set up windows to do this depends on what version of windows is on each computer, if both XP it is very very easy.
If you have a firewall eg. zone alarm, you need to reconfigure that to allow access.